Methodology and method and apparatus for signaling with capacity optimized constellations

ABSTRACT

Communication systems having transmitter, includes a coder configured to receive user bits and output encoded bits at an expanded output encoded bit rate, a mapper configured to map encoded bits to symbols in a symbol constellation, a modulator configured to generate a signal for transmission via the communication channel using symbols generated by the mapper. In addition, the receiver includes a demodulator configured to demodulate the received signal via the communication channel, a demapper configured to estimate likelihoods from the demodulated signal, a decoder that is configured to estimate decoded bits from the likelihoods generated by the demapper. Furthermore, the symbol constellation is a capacity optimized geometrically spaced symbol constellation that provides a given capacity at a reduced signal-to-noise ratio compared to a signal constellation that maximizes d min .

CROSS-REFERENCE TO RELATED APPLICATIONS

The present invention claims priority to U.S. Provisional ApplicationSer. No. 60/933,319 entitled “New Constellations for CommunicationsSignaling: Design Methodology and Method and Apparatus for the NewSignaling Scheme” to Barsoum et al., filed Jun. 5, 2007, the disclosureof which is expressly incorporated by reference herein in its entirety.

STATEMENT OF FEDERALLY SPONSORED RESEARCH

This invention was made with Government support under contractNAS7-03001 awarded by NASA. The Government has certain rights in thisinvention.

BACKGROUND

The present invention generally relates to bandwidth and/or powerefficient digital transmission systems and more specifically to the useof unequally spaced constellations having increased capacity.

The term “constellation” is used to describe the possible symbols thatcan be transmitted by a typical digital communication system. A receiverattempts to detect the symbols that were transmitted by mapping areceived signal to the constellation. The minimum distance (d_(min))between constellation points is indicative of the capacity of aconstellation at high signal-to-noise ratios (SNRs). Therefore,constellations used in many communication systems are designed tomaximize d_(min). Increasing the dimensionality of a constellationallows larger minimum distance for constant constellation energy perdimension. Therefore, a number of multi-dimensional constellations withgood minimum distance properties have been designed.

Communication systems have a theoretical maximum capacity, which isknown as the Shannon limit. Many communication systems attempt to usecodes to increase the capacity of a communication channel. Significantcoding gains have been achieved using coding techniques such as turbocodes and LDPC codes. The coding gains achievable using any codingtechnique are limited by the constellation of the communication system.The Shannon limit can be thought of as being based upon a theoreticalconstellation known as a Gaussian distribution, which is an infiniteconstellation where symbols at the center of the constellation aretransmitted more frequently than symbols at the edge of theconstellation. Practical constellations are finite and transmit symbolswith equal likelihoods, and therefore have capacities that are less thanthe Gaussian capacity. The capacity of a constellation is thought torepresent a limit on the gains that can be achieved using coding whenusing that constellation.

Prior attempts have been made to develop unequally spacedconstellations. For example, a system has been proposed that usesunequally spaced constellations that are optimized to minimize the errorrate of an uncoded system. Another proposed system uses a constellationwith equiprobable but unequally spaced symbols in an attempts to mimic aGaussian distribution.

Other approaches increases the dimensionality of a constellation orselect a new symbol to be transmitted taking into considerationpreviously transmitted symbols. However, these constellation were stilldesigned based on a minimum distance criteria.

SUMMARY OF THE INVENTION

Systems and methods are described for constructing a modulation suchthat the constrained capacity between a transmitter and a receiverapproaches the Gaussian channel capacity limit first described byShannon [ref Shannon 1948]. Traditional communications systems employmodulations that leave a significant gap to Shannon Gaussian capacity.The modulations of the present invention reduce, and in some cases,nearly eliminate this gap. The invention does not require speciallydesigned coding mechanisms that tend to transmit some points of amodulation more frequently than others but rather provides a method forlocating points (in a one or multiple dimensional space) in order tomaximize capacity between the input and output of a bit or symbol mapperand demapper respectively. Practical application of the method allowssystems to transmit data at a given rate for less power or to transmitdata at a higher rate for the same amount of power.

One embodiment of the invention includes a transmitter configured totransmit signals to a receiver via a communication channel, wherein thetransmitter, includes a coder configured to receive user bits and outputencoded bits at an expanded output encoded bit rate, a mapper configuredto map encoded bits to symbols in a symbol constellation, a modulatorconfigured to generate a signal for transmission via the communicationchannel using symbols generated by the mapper. In addition, the receiverincludes a demodulator configured to demodulate the received signal viathe communication channel, a demapper configured to estimate likelihoodsfrom the demodulated signal, a decoder that is configured to estimatedecoded bits from the likelihoods generated by the demapper.Furthermore, the symbol constellation is a capacity optimizedgeometrically spaced symbol constellation that provides a given capacityat a reduced signal-to-noise ratio compared to a signal constellationthat maximizes d_(min).

A further embodiment of the invention includes encoding the bits of userinformation using a coding scheme, mapping the encoded bits of userinformation to a symbol constellation, wherein the symbol constellationis a capacity optimized geometrically spaced symbol constellation thatprovides a given capacity at a reduced signal-to-noise ratio compared toa signal constellation that maximizes d_(min), modulating the symbols inaccordance with a modulation scheme, transmitting the modulated signalvia the communication channel, receiving a modulated signal,demodulating the modulated signal in accordance with the modulationscheme, demapping the demodulated signal using the geometrically shapedsignal constellation to produce likelihoods, and decoding thelikelihoods to obtain an estimate of the decoded bits.

Another embodiment of the invention includes selecting an appropriateconstellation size and a desired capacity per dimension, estimating aninitial SNR at which the system is likely to operate, and iterativelyoptimizing the location of the points of the constellation to maximize acapacity measure until a predetermined improvement in the SNRperformance of the constellation relative to a constellation thatmaximizes d_(min) has been achieved.

A still further embodiment of the invention includes selecting anappropriate constellation size and a desired capacity per dimension,estimating an initial SNR at which the system is likely to operate, anditeratively optimizing the location of the points of the constellationto maximize a capacity measure until a predetermined improvement in theSNR performance of the constellation relative to a constellation thatmaximizes d_(min) has been achieved.

Still another embodiment of the invention includes selecting anappropriate constellation size and a desired SNR, and optimizing thelocation of the points of the constellation to maximize a capacitymeasure of the constellation.

A yet further embodiment of the invention includes obtaining ageometrically shaped PAM constellation with a constellation size that isthe square root of said given constellation size, where thegeometrically shaped PAM constellation has a capacity greater than thatof a PAM constellation that maximizes d_(min), creating anorthogonalized PAM constellation using the geometrically shaped PAMconstellation, and combining the geometrically shaped PAM constellationand the orthogonalized PAM constellation to produce a geometricallyshaped QAM constellation.

Another further embodiment of the invention includes transmittinginformation over a channel using a geometrically shaped symbolconstellation, and modifying the location of points within thegeometrically shaped symbol constellation to change the target user datarate.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a conceptual illustration of a communication system inaccordance with an embodiment of the invention.

FIG. 2 is a conceptual illustration of a transmitter in accordance withan embodiment of the invention.

FIG. 3 is a conceptual illustration of a receiver in accordance with anembodiment of the invention.

FIG. 4 a is a conceptual illustration of the joint capacity of achannel.

FIG. 4 b is a conceptual illustration of the parallel decoding capacityof a channel.

FIG. 5 is a flow chart showing a process for obtaining a constellationoptimized for capacity for use in a communication system having a fixedcode rate and modulation scheme in accordance with an embodiment of theinvention.

FIG. 6 a is a chart showing a comparison of Gaussian capacity and PDcapacity for traditional PAM-2,4,8,16,32.

FIG. 6 b is a chart showing a comparison between Gaussian capacity andjoint capacity for traditional PAM-2,4,8,16,32.

FIG. 7 is a chart showing the SNR gap to Gaussian capacity for the PDcapacity and joint capacity of traditional PAM-2,4,8,16,32constellations.

FIG. 8 a is a chart comparing the SNR gap to Gaussian capacity of the PDcapacity for traditional and optimized PAM-2,4,8,16,32 constellations.

FIG. 8 b is a chart comparing the SNR gap to Gaussian capacity of thejoint capacity for traditional and optimized PAM-2,4,8,16,32constellations.

FIG. 9 is a chart showing Frame Error Rate performance of traditionaland PD capacity optimized PAM-32 constellations in simulations involvingseveral different length LDPC codes.

FIGS. 10 a-10 d are locus plots showing the location of constellationpoints of a PAM-4 constellation optimized for PD capacity and jointcapacity versus user bit rate per dimension and versus SNR.

FIGS. 11 a and 11 b are design tables of PD capacity and joint capacityoptimized PAM-4 constellations in accordance with embodiments of theinvention.

FIGS. 12 a-12 d are locus plots showing the location of constellationpoints of a PAM-8 constellation optimized for PD capacity and jointcapacity versus user bit rate per dimension and versus SNR.

FIGS. 13 a and 13 b are design tables of PD capacity and joint capacityoptimized PAM-8 constellations in accordance with embodiments of theinvention.

FIGS. 14 a-14 d are locus plots showing the location of constellationpoints of a PAM-16 constellation optimized for PD capacity and jointcapacity versus user bit rate per dimension and versus SNR.

FIGS. 15 a and 15 b are design tables of PD capacity and joint capacityoptimized PAM-16 constellations in accordance with embodiments of theinvention.

FIGS. 16 a-16 d are locus plots showing the location of constellationpoints of a PAM-32 constellation optimized for PD capacity and jointcapacity versus user bit rate per dimension and versus SNR.

FIGS. 17 a and 17 b are design tables of PD capacity and joint capacityoptimized PAM-32 constellations in accordance with embodiments of theinvention.

FIG. 18 is a chart showing the SNR gap to Gaussian capacity fortraditional and capacity optimized PSK constellations.

FIG. 19 is a chart showing the location of constellation points of PDcapacity optimized PSK-32 constellations.

FIG. 20 is a series of PSK-32 constellations optimized for PD capacityat different SNRs in accordance with embodiments of the invention.

FIG. 21 illustrates a QAM-64 constructed from orthogonal Cartesianproduct of two PD optimized PAM-8 constellations in accordance with anembodiment of the invention.

FIGS. 22 a and 22 b are locus plots showing the location ofconstellation points of a PAM-4 constellation optimized for PD capacityover a fading channel versus user bit rate per dimension and versus SNR.

FIGS. 23 a and 23 b are locus plots showing the location ofconstellation points of a PAM-8 constellation optimized for PD capacityover a fading channel versus user bit rate per dimension and versus SNR.

FIGS. 24 a and 24 b are locus plots showing the location ofconstellation points of a PAM-16 constellation optimized for PD capacityover a fading channel versus user bit rate per dimension and versus SNR.

DETAILED DESCRIPTION OF THE INVENTION

Turning now to the drawings, communication systems in accordance withembodiments of the invention are described that use signalconstellations, which have unequally spaced (i.e. ‘geometrically’shaped) points. In several embodiments, the locations of geometricallyshaped points are designed to provide a given capacity measure at areduced signal-to-noise ratio (SNR) compared to the SNR required by aconstellation that maximizes d_(min). In many embodiments, theconstellations are selected to provide increased capacity at apredetermined range of channel signal-to-noise ratios (SNR). Capacitymeasures that can be used in the selection of the location ofconstellation points include, but are not limited to, parallel decode(PD) capacity and joint capacity.

In many embodiments, the communication systems utilize capacityapproaching codes including, but not limited to, LDPC and Turbo codes.As is discussed further below, direct optimization of the constellationpoints of a communication system utilizing a capacity approachingchannel code, can yield different constellations depending on the SNRfor which they are optimized. Therefore, the same constellation isunlikely to achieve the same coding gains applied across all code rates;that is, the same constellation will not enable the best possibleperformance across all rates. In many instances, a constellation at onecode rate can achieve gains that cannot be achieved at another coderate. Processes for selecting capacity optimized constellations toachieve increased coding gains based upon a specific coding rate inaccordance with embodiments of the invention are described below. In anumber of embodiments, the communication systems can adapt location ofpoints in a constellation in response to channel conditions, changes incode rate and/or to change the target user data rate.

Communication Systems

A communication system in accordance with an embodiment of the inventionis shown in FIG. 1. The communication system 10 includes a source 12that provides user bits to a transmitter 14. The transmitter transmitssymbols over a channel to a receiver 16 using a predetermined modulationscheme. The receiver uses knowledge of the modulation scheme, to decodethe signal received from the transmitter. The decoded bits are providedto a sink device that is connected to the receiver.

A transmitter in accordance with an embodiment of the invention is shownin FIG. 2. The transmitter 14 includes a coder 20 that receives userbits from a source and encodes the bits in accordance with apredetermined coding scheme. In a number of embodiments, a capacityapproaching code such as a turbo code or a LDPC code is used. In otherembodiments, other coding schemes can be used to providing a coding gainwithin the communication system. A mapper 22 is connected to the coder.The mapper maps the bits output by the coder to a symbol within ageometrically distributed signal constellation stored within the mapper.The mapper provides the symbols to a modulator 24, which modulates thesymbols for transmission via the channel.

A receiver in accordance with an embodiment of the invention isillustrated in FIG. 3. The receiver 16 includes a demodulator 30 thatdemodulates a signal received via the channel to obtain symbol or bitlikelihoods. The demapper uses knowledge of the geometrically shapedsymbol constellation used by the transmitter to determine theselikelihoods. The demapper 32 provides the likelihoods to a decoder 34that decodes the encoded bit stream to provide a sequence of receivedbits to a sink.

Geometrically Shaped Constellations

Transmitters and receivers in accordance with embodiments of theinvention utilize geometrically shaped symbol constellations. In severalembodiments, a geometrically shaped symbol constellation is used thatoptimizes the capacity of the constellation. Various geometricallyshaped symbol constellations that can be used in accordance withembodiments of the invention, techniques for deriving geometricallyshaped symbol constellations are described below.

Selection of a Geometrically Shaped Constellations

Selection of a geometrically shaped constellation for use in acommunication system in accordance with an embodiment of the inventioncan depend upon a variety of factors including whether the code rate isfixed. In many embodiments, a geometrically shaped constellation is usedto replace a conventional constellation (i.e. a constellation maximizedfor d_(min)) in the mapper of transmitters and the demapper of receiverswithin a communication system. Upgrading a communication system involvesselection of a constellation and in many instances the upgrade can beachieved via a simple firmware upgrade. In other embodiments, ageometrically shaped constellation is selected in conjunction with acode rate to meet specific performance requirements, which can forexample include such factors as a specified bit rate, a maximum transmitpower. Processes for selecting a geometric constellation when upgradingexisting communication systems and when designing new communicationsystems are discussed further below.

Upgrading Existing Communication Systems

A geometrically shaped constellation that provides a capacity, which isgreater than the capacity of a constellation maximized for d_(min), canbe used in place of a conventional constellation in a communicationsystem in accordance with embodiments of the invention. In manyinstances, the substitution of the geometrically shaped constellationcan be achieved by a firmware or software upgrade of the transmittersand receivers within the communication system. Not all geometricallyshaped constellations have greater capacity than that of a constellationmaximized for d_(min). One approach to selecting a geometrically shapedconstellation having a greater capacity than that of a constellationmaximized for d_(min) is to optimize the shape of the constellation withrespect to a measure of the capacity of the constellation for a givenSNR. Capacity measures that can be used in the optimization process caninclude, but are not limited to, joint capacity or parallel decodingcapacity.

Joint Capacity and Parallel Decoding Capacity

A constellation can be parameterized by the total number ofconstellation points, M, and the number of real dimensions, N_(dim). Insystems where there are no belief propagation iterations between thedecoder and the constellation demapper, the constellation demapper canbe thought of as part of the channel. A diagram conceptuallyillustrating the portions of a communication system that can beconsidered part of the channel for the purpose of determining PDcapacity is shown in FIG. 4 a. The portions of the communication systemthat are considered part of the channel are indicated by the ghost line40. The capacity of the channel defined as such is the parallel decoding(PD) capacity, given by:

$C_{PD} = {\sum\limits_{i = 0}^{l - 1}{I\left( {X_{i};Y} \right)}}$where X_(i) is the ith bit of the l-bits transmitted symbol, and Y isthe received symbol, and I(A;B) denotes the mutual information betweenrandom variables A and B.

Expressed another way, the PD capacity of a channel can be viewed interms of the mutual information between the output bits of the encoder(such as an LDPC encoder) at the transmitter and the likelihoodscomputed by the demapper at the receiver. The PD capacity is influencedby both the placement of points within the constellation and by thelabeling assignments.

With belief propagation iterations between the demapper and the decoder,the demapper can no longer be viewed as part of the channel, and thejoint capacity of the constellation becomes the tightest known bound onthe system performance. A diagram conceptually illustrating the portionsof a communication system that are considered part of the channel forthe purpose of determining the joint capacity of a constellation isshown in FIG. 4 b. The portions of the communication system that areconsidered part of the channel are indicated by the ghost line 42. Thejoint capacity of the channel is given by:C _(JOINT) =I(X;Y)

Joint capacity is a description of the achievable capacity between theinput of the mapper on the transmit side of the link and the output ofthe channel (including for example AWGN and Fading channels). Practicalsystems must often ‘demap’ channel observations prior to decoding. Ingeneral, the step causes some loss of capacity. In fact it can be proventhat C_(G)≧C_(JOINT)≧C_(PD). That is, C_(JOINT) upper bounds thecapacity achievable by C_(PD). The methods of the present invention aremotivated by considering the fact that practical limits to a givencommunication system capacity are limited by C_(JOINT) and C_(PD). Inseveral embodiments of the invention, geometrically shapedconstellations are selected that maximize these measures.

Selecting a Constellation Having an Optimal Capacity

Geometrically shaped constellations in accordance with embodiments ofthe invention can be designed to optimize capacity measures including,but not limited to PD capacity or joint capacity. A process forselecting the points, and potentially the labeling, of a geometricallyshaped constellation for use in a communication system having a fixedcode rate in accordance with an embodiment of the invention is shown inFIG. 5. The process 50 commences with the selection (52) of anappropriate constellation size M and a desired capacity per dimension η.In the illustrated embodiment, the process involves a check (52) toensure that the constellation size can support the desired capacity. Inthe event that the constellation size could support the desiredcapacity, then the process iteratively optimizes the M-ary constellationfor the specified capacity. Optimizing a constellation for a specifiedcapacity often involves an iterative process, because the optimalconstellation depends upon the SNR at which the communication systemoperates. The SNR for the optimal constellation to give a requiredcapacity is not known a priori. Throughout the description of thepresent invention SNR is defined as the ratio of the averageconstellation energy per dimension to the average noise energy perdimension. In most cases the capacity can be set to equal the targetuser bit rate per symbol per dimension. In some cases adding someimplementation margin on top of the target user bit rate could result ina practical system that can provide the required user rate at a lowerrate. The margin is code dependent. The following procedure could beused to determine the target capacity that includes some margin on topof the user rate. First, the code (e.g. LDPC or Turbo) can be simulatedin conjunction with a conventional equally spaced constellation. Second,from the simulation results the actual SNR of operation at the requirederror rate can be found. Third, the capacity of the conventionalconstellation at that SNR can be computed. Finally, a geometricallyshaped constellation can be optimized for that capacity.

In the illustrated embodiment, the iterative optimization loop involvesselecting an initial estimate of the SNR at which the system is likelyto operate (i.e. SNR_(in)). In several embodiments the initial estimateis the SNR required using a conventional constellation. In otherembodiments, other techniques can be used for selecting the initial SNR.An M-ary constellation is then obtained by optimizing (56) theconstellation to maximize a selected capacity measure at the initialSNR_(in) estimate. Various techniques for obtaining an optimizedconstellation for a given SNR estimate are discussed below.

The SNR at which the optimized M-ary constellation provides the desiredcapacity per dimension η (SNR_(out)) is determined (57). A determination(58) is made as to whether the SNR_(out) and SNR_(in) have converged. Inthe illustrated embodiment convergence is indicated by SNR_(out)equaling SNR_(in). In a number of embodiments, convergence can bedetermined based upon the difference between SNR_(out) and SNR_(in)being less than a predetermined threshold. When SNR_(out) and SNR_(in)have not converged, the process performs another iteration selectingSNR_(out) as the new SNR_(in) (55). When SNR_(out) and SNR_(in) haveconverged, the capacity measure of the constellation has been optimized.As is explained in more detail below, capacity optimized constellationat low SNRs are geometrically shaped constellations that can achievesignificantly higher performance gains (measured as reduction in minimumrequired SNR) than constellations that maximize d_(min).

The process illustrated in FIG. 5 can maximize PD capacity or jointcapacity of an M-ary constellation for a given SNR. Although the processillustrated in FIG. 5 shows selecting an M-ary constellation optimizedfor capacity, a similar process could be used that terminates upongeneration of an M-ary constellation where the SNR gap to Gaussiancapacity at a given capacity is a predetermined margin lower than theSNR gap of a conventional constellation, for example 0.5 db.Alternatively, other processes that identify M-ary constellations havingcapacity greater than the capacity of a conventional constellation canbe used in accordance with embodiments of the invention. A geometricallyshaped constellation in accordance with embodiments of the invention canachieve greater capacity than the capacity of a constellation thatmaximizes d_(min) without having the optimal capacity for the SNR rangewithin which the communication system operates.

We note that constellations designed to maximize joint capacity may alsobe particularly well suited to codes with symbols over GF(q), or withmulti-stage decoding. Conversely constellations optimized for PDcapacity could be better suited to the more common case of codes withsymbols over GF(2)

Optimizing the capacity of an M-ary constellation at a given SNR

Processes for obtaining a capacity optimized constellation often involvedetermining the optimum location for the points of an M-aryconstellation at a given SNR. An optimization process, such as theoptimization process 56 shown in FIG. 5, typically involvesunconstrained or constrained non-linear optimization. Possible objectivefunctions to be maximized are the Joint or PD capacity functions. Thesefunctions may be targeted to channels including but not limited toAdditive White Gaussian Noise (AWGN) or Rayleigh fading channels. Theoptimization process gives the location of each constellation pointidentified by its symbol labeling. In the case where the objective isjoint capacity, point bit labelings are irrelevant meaning that changingthe bit labelings doesn't change the joint capacity as long as the setof point locations remains unchanged.

The optimization process typically finds the constellation that givesthe largest PD capacity or joint capacity at a given SNR. Theoptimization process itself often involves an iterative numericalprocess that among other things considers several constellations andselects the constellation that gives the highest capacity at a givenSNR. In other embodiments, the constellation that requires the least SNRto give a required PD capacity or joint capacity can also be found. Thisrequires running the optimization process iteratively as shown in FIG.5.

Optimization constraints on the constellation point locations mayinclude, but are not limited to, lower and upper bounds on pointlocation, peak to average power of the resulting constellation, and zeromean in the resulting constellation. It can be easily shown that aglobally optimal constellation will have zero mean (no DC component).Explicit inclusion of a zero mean constraint helps the optimizationroutine to converge more rapidly. Except for cases where exhaustivesearch of all combinations of point locations and labelings is possibleit will not necessarily always be the case that solutions are provablyglobally optimal. In cases where exhaustive search is possible, thesolution provided by the non-linear optimizer is in fact globallyoptimal.

The processes described above provide examples of the manner in which ageometrically shaped constellation having an increased capacity relativeto a conventional capacity can be obtained for use in a communicationsystem having a fixed code rate and modulation scheme. The actual gainsachievable using constellations that are optimized for capacity comparedto conventional constellations that maximize d_(min) are consideredbelow.

Gains Achieved by Optimized Geometrically Spaced Constellations

The ultimate theoretical capacity achievable by any communication methodis thought to be the Gaussian capacity, C_(G) which is defined as:

$C_{G} = {\frac{1}{2}{\log_{2}\left( {1 + {S\; N\; R}} \right)}}$

Where signal-to-noise (SNR) is the ratio of expected signal power toexpected noise power. The gap that remains between the capacity of aconstellation and C_(G) can be considered a measure of the quality of agiven constellation design.

The gap in capacity between a conventional modulation scheme incombination with a theoretically optimal coder can be observed withreference to FIGS. 6 a and 6 b. FIG. 6 a includes a chart 60 showing acomparison between Gaussian capacity and the PD capacity of conventionalPAM-2, 4, 8, 16, and 32 constellations that maximize d_(min), Gaps 62exist between the plot of Gaussian capacity and the PD capacity of thevarious PAM constellations. FIG. 6 b includes a chart 64 showing acomparison between Gaussian capacity and the joint capacity ofconventional PAM-2, 4, 8, 16, and 32 constellations that maximized_(min), Gaps 66 exist between the plot of Gaussian capacity and thejoint capacity of the various PAM constellations. These gaps in capacityrepresent the extent to which conventional PAM constellations fall shortof obtaining the ultimate theoretical capacity i.e. the Gaussiancapacity.

In order to gain a better view of the differences between the curvesshown in FIGS. 6 a and 6 b at points close to the Gaussian capacity, theSNR gap to Gaussian capacity for different values of capacity for eachconstellation are plotted in FIG. 7. It is interesting to note from thechart 70 in FIG. 7 that (unlike the joint capacity) at the same SNR, thePD capacity does not necessarily increase with the number ofconstellation points. As is discussed further below, this is not thecase with PAM constellations optimized for PD capacity.

FIGS. 8 a and 8 b summarize performance of constellations for PAM-4, 8,16, and 32 optimized for PD capacity and joint capacity (it should benoted that BPSK is the optimal PAM-2 constellation at all code rates).The constellations are optimized for PD capacity and joint capacity fordifferent target user bits per dimension (i.e. code rates). Theoptimized constellations are different depending on the target user bitsper dimension, and also depending on whether they have been designed tomaximize the PD capacity or the joint capacity. All the PD optimized PAMconstellations are labeled using a gray labeling which is not always thebinary reflective gray labeling. It should be noted that not all graylabels achieve the maximum possible PD capacity even given the freedomto place the constellation points anywhere on the real line. FIG. 8 ashows the SNR gap for each constellation optimized for PD capacity. FIG.8 b shows the SNR gap to Gaussian capacity for each constellationoptimized for joint capacity. Again, it should be emphasized that each‘+’ on the plot represents a different constellation.

Referring to FIG. 8 a, the coding gain achieved using a constellationoptimized for PD capacity can be appreciated by comparing the SNR gap ata user bit rate per dimension of 2.5 bits for PAM-32. A user bit rateper dimension of 2.5 bits for a system transmitting 5 bits per symbolconstitutes a code rate of ½. At that code rate the constellationoptimized for PD capacity provides an additional coding gain ofapproximately 1.5 dB when compared to the conventional PAM-32constellation.

The SNR gains that can be achieved using constellations that areoptimized for PD capacity can be verified through simulation. Theresults of a simulation conducted using a rate ½ LDPC code inconjunction with a conventional PAM-32 constellation and in conjunctionwith a PAM-32 constellation optimized for PD capacity are illustrated inFIG. 9. A chart 90 includes plots of Frame Error Rate performance of thedifferent constellations with respect to SNR and using different lengthcodes (i.e. k=4,096 and k=16,384). Irrespective of the code that isused, the constellation optimized for PD capacity achieves a gain ofapproximately 1.3 dB, which closely approaches the gain predicted fromFIG. 8 a.

Capacity Optimized Pam Constellations

Using the processes outlined above, locus plots of PAM constellationsoptimized for capacity can be generated that show the location of pointswithin PAM constellations versus SNR. Locus plots of PAM-4, 8, 16, and32 constellations optimized for PD capacity and joint capacity andcorresponding design tables at various typical user bit rates perdimension are illustrated in FIGS. 10 a-17 b. The locus plots and designtables show PAM-4,8,16,32 constellation point locations and labelingsfrom low to high SNR corresponding to a range of low to high spectralefficiency.

In FIG. 10 a, a locus plot 100 shows the location of the points of PAM-4constellations optimized for Joint capacity plotted against achievedcapacity. A similar locus plot 105 showing the location of the points ofJoint capacity optimized PAM-4 constellations plotted against SNR isincluded in FIG. 10 b. In FIG. 10 c. the location of points for PAM-4optimized for PD capacity is plotted against achievable capacity and inFIG. 10 d the location of points for PAM-4 for PD capacity is plottedagainst SNR. At low SNRs, the PD capacity optimized PAM-4 constellationshave only 2 unique points, while the Joint optimized constellations have3. As SNR is increased, each optimization eventually provides 4 uniquepoints. This phenomenon is explicitly described in FIG. 11 a and FIG. 11b where vertical slices of FIGS. 10 ab and 10 cd are captured in tablesdescribing some PAM-4 constellations designs of interest. The SNR slicesselected represent designs that achieve capacities={0.5, 0.75, 1.0,1.25, 1.5} bits per symbol (bps). Given that PAM-4 can provide at mostlog₂(4)=2 bps, these design points represent systems with informationcode rates R={¼, ⅜, ½, ⅝, ¾} respectively.

FIGS. 12 ab and 12 cd present locus plots of PD capacity and jointcapacity optimized PAM-8 constellation points versus achievable capacityand SNR. FIGS. 13 a and 13 b provide slices from these plots at SNRscorresponding to achievable capacities η={0.5, 1.0, 1.5, 2.0, 2.5} bps.Each of these slices correspond to systems with code rate R=ηbps/log₂(8), resulting in R={⅙, ⅓, ½, ⅔, ⅚}. As an example of therelative performance of the constellations in these tables, considerFIG. 13 b which shows a PD capacity optimized PAM-8 constellationoptimized for SNR=9.00 dB, or 1.5 bps. We next examine the plot providedin FIG. 8 a and see that the gap of the optimized constellation to theultimate, Gaussian, capacity (CG) is approximately 0.5 dB. At the samespectral efficiency, the gap of the traditional PAM-8 constellation isapproximately 1.0 dB. The advantage of the optimized constellation is0.5 dB for the same rate (in this case R=½). This gain can be obtainedby only changing the mapper and demapper in the communication system andleaving all other blocks the same.

Similar information is presented in FIGS. 14 abcd, and 15 ab whichprovide loci plots and design tables for PAM-16 PD capacity and jointcapacity optimized constellations. Likewise FIGS. 16 abcd, 17 ab provideloci plots and design tables for PAM-32 PD capacity and joint capacityoptimized constellations.

Capacity Optimized PSK Constellations

Traditional phase shift keyed (PSK) constellations are already quiteoptimal. This can be seen in the chart 180 comparing the SNR gaps oftradition PSK with capacity optimized PSK constellations shown in FIG.18 where the gap between PD capacity and Gaussian capacity is plottedfor traditional PSK-4,8,16,32 and for PD capacity optimizedPSK-4,8,16,32.

The locus plot of PD optimized PSK-32 points across SNR is shown in FIG.19, which actually characterizes all PSKs with spectral efficiency η≦5.This can be seen in FIG. 20. Note that at low SNR (0.4 dB) the optimalPSK-32 design is the same as traditional PSK-4, at SNR=8.4 dB optimalPSK-32 is the same as traditional PSK-8, at SNR=14.8 dB optimal PSK-32is the same as traditional PSK-16, and finally at SNRs greater than 20.4dB optimized PSK-32 is the same as traditional PSK-32. There are SNRsbetween these discrete points (for instance SNR=2 and 15. dB) for whichoptimized PSK-32 provides superior PD capacity when compared totraditional PSK constellations.

We note now that the locus of points for PD optimized PSK-32 in FIG. 19in conjunction with the gap to Gaussian capacity curve for optimizedPSK-32 in FIG. 18 implies a potential design methodology. Specifically,the designer could achieve performance equivalent or better than thatenabled by traditional PSK-4,8,16 by using only the optimized PSK-32 inconjunction with a single tuning parameter that controlled where theconstellation points should be selected from on the locus of FIG. 19.Such an approach would couple a highly rate adaptive channel code thatcould vary its rate, for instance, rate ⅘ to achieve and overall (codeplus optimized PSK-32 modulation) spectral efficiency of 4 bits persymbol, down to ⅕ to achieve an overall spectral efficiency of 1 bit persymbol. Such an adaptive modulation and coding system could essentiallyperform on the optimal continuum represented by the rightmost contour ofFIG. 18.

Adaptive Rate Design

In the previous example spectrally adaptive use of PSK-32 was described.Techniques similar to this can be applied for other capacity optimizedconstellations across the link between a transmitter and receiver. Forinstance, in the case where a system implements quality of service it ispossible to instruct a transmitter to increase or decrease spectralefficiency on demand. In the context of the current invention a capacityoptimized constellation designed precisely for the target spectralefficiency can be loaded into the transmit mapper in conjunction with acode rate selection that meets the end user rate goal. When such amodulation/code rate change occurred a message could propagated to thereceiver so that the receiver, in anticipation of the change, couldselect a demapper/decoder configuration in order to match the newtransmit-side configuration.

Conversely, the receiver could implement a quality of performance basedoptimized constellation/code rate pair control mechanism. Such anapproach would include some form of receiver quality measure. This couldbe the receiver's estimate of SNR or bit error rate. Take for examplethe case where bit error rate was above some acceptable threshold. Inthis case, via a backchannel, the receiver could request that thetransmitter lower the spectral efficiency of the link by swapping to analternate capacity optimized constellation/code rate pair in the coderand mapper modules and then signaling the receiver to swap in thecomplementary pairing in the demapper/decoder modules.

Geometrically Shaped QAM Constellations

Quadrature amplitude modulation (QAM) constellations can be constructedby orthogonalizing PAM constellations into QAM inphase and quadraturecomponents. Constellations constructed in this way can be attractive inmany applications because they have low-complexity demappers.

In FIG. 21 we provide an example of a Quadrature Amplitude Modulationconstellation constructed from a Pulse Amplitude Modulationconstellation. The illustrated embodiment was constructed using a PAM-8constellation optimized for PD capacity at user bit rate per dimensionof 1.5 bits (corresponds to an SNR of 9.0 dB) (see FIG. 13 b). Thelabel-point pairs in this PAM-8 constellation are {(000, −1.72), (001,−0.81), (010, 1.72), (011, −0.62), (100, 0.62), (101, 0.02), (110,0.81), (111, −0.02)}. Examination of FIG. 21 shows that the QAMconstellation construction is achieved by replicating a complete set ofPAM-8 points in the quadrature dimension for each of the 8 PAM-8 pointsin the in-phase dimension. Labeling is achieved by assigning the PAM-8labels to the LSB range on the in-phase dimension and to the MSB rangeon the quadrature dimension. The resulting 8×8 outer product forms ahighly structured QAM-64 for which very low-complexity de-mappers can beconstructed. Due to the orthogonality of the in-phase and quadraturecomponents the capacity characteristics of the resulting QAM-64constellation are identical to that of the PAM-8 constellation on aper-dimension basis.

N-Dimensional Constellation Optimization

Rather than designing constellations in 1-D (PAM for instance) and thenextending to 2-D (QAM), it is possible to take direct advantage in theoptimization step of the additional degree of freedom presented by anextra spatial dimension. In general it is possible to designN-dimensional constellations and associated labelings. The complexity ofthe optimization step grows exponentially in the number of dimensions asdoes the complexity of the resulting receiver de-mapper. Suchconstructions constitute embodiments of the invention and simply requiremore ‘run-time’ to produce.

Capacity Optimized Constellations For Fading Channels

Similar processes to those outlined above can be used to design capacityoptimized constellations for fading channels in accordance withembodiments of the invention. The processes are essentially the samewith the exception that the manner in which capacity is calculated ismodified to account for the fading channel. A fading channel can bedescribed using the following equation:Y=α(t)·X+Nwhere X is the transmitted signal, N is an additive white Gaussian noisesignal and a(t) is the fading distribution, which is a function of time.

In the case of a fading channel, the instantaneous SNR at the receiverchanges according to a fading distribution. The fading distribution isRayleigh and has the property that the average SNR of the system remainsthe same as in the case of the AWGN channel, E[X²]/E[N²]. Therefore, thecapacity of the fading channel can be computed by taking the expectationof AWGN capacity, at a given average SNR, over the Rayleigh fadingdistribution of a that drives the distribution of the instantaneous SNR.

Many fading channels follow a Rayleigh distribution. FIGS. 22 a-24 b arelocus plots of PAM-4, 8, and 16 constellations that have been optimizedfor PD capacity on a Rayleigh fading channel. Locus plots versus userbit rate per dimension and versus SNR are provided. Similar processescan be used to obtain capacity optimized constellations that areoptimized using other capacity measures, such as joint capacity, and/orusing different modulation schemes.

1. A digital communication system, comprising: a transmitter configuredto transmit signals to a receiver via a communication channel; whereinthe transmitter, comprises: a coder configured to receive user bits andoutput encoded bits at an expanded output encoded bit rate; a mapperconfigured to map encoded bits to symbols in a symbol constellation; amodulator configured to generate a signal for transmission via thecommunication channel using symbols generated by the mapper; wherein thereceiver, comprises: a demodulator configured to demodulate the receivedsignal via the communication channel; a demapper configured to estimatelikelihoods from the demodulated signal; a decoder that is configured toestimate decoded bits from the likelihoods generated by the demapper;and wherein the symbol constellation is a capacity optimizedgeometrically spaced symbol constellation that provides a given capacityat a reduced signal-to-noise ratio compared to a signal constellationthat maximizes d_(min).
 2. The communication system of claim 1, whereinthe geometrically spaced symbol constellation is capacity optimizedsubject to additional constraints.
 3. The communication system of claim1, wherein the geometrically shaped symbol constellation is optimizedfor capacity using parallel decode capacity.
 4. The communication systemof claim 1, wherein the geometrically shaped symbol constellation isoptimized for capacity using joint capacity.
 5. The communication systemof claim 1, wherein the code is a Turbo code.
 6. The communicationsystem of claim 1, wherein the code is a LDPC code.
 7. The communicationsystem of claim 1, wherein the modulation scheme is a PAM scheme.
 8. Thecommunication system of claim 1, wherein the modulation scheme is a PSKscheme.
 9. The communication system of claim 1, wherein the modulationscheme is a QAM scheme.
 10. The communication system of claim 1, whereinthe modulation scheme is a multidimensional modulation scheme.
 11. Thecommunication system of claim 1, wherein the channel is an AWGN channel.12. The communication system of claim 1, wherein the channel is a fadingchannel.
 13. The communication system of claim 1, wherein theconstellations points of the geometrically spaced symbol constellationare variable depending on the user rate.
 14. The communication system ofclaim 13, wherein: the transmitter is configured to modify the rate ofthe code in order to change the user bit rate of the communicationsystem; the transmitter is configured to move the constellation pointsof the symbol constellation used by the mapper based upon the code rateof the coder; the receiver is configured to receive informationconcerning code rate changes from the transmitter; and the receiver isconfigured to modify the decoder and the constellation used by thedemapper in response to receipt of a notification of a change of userbit rate.
 15. The communication system of claim 14, wherein: thereceiver is configured to acknowledge receipt of a modification to thecode rate and the constellation of the communication system; and thetransmitter is configured to commence transmitting using the new coderate and symbol constellation upon receipt of the acknowledgement fromthe receiver.
 16. The communication system of claim 15, wherein: abackchannel communication link exists between the transmitter and thereceiver; the receiver is configured to monitor receiver performance andinitiate a change in code rate in response to the observed receiverperformance; the receiver is configured to transmit to the transmittervia the backchannel a change in user bit rate; and both the mapper andthe demapper are configured to move the constellation points of thegeometrically shaped symbol constellation based upon the change in userbit rate.
 17. The communication system of claim 1, wherein theconstellations points of the geometrically spaced symbol constellationare variable depending on the channel.
 18. The communication system ofclaim 17, wherein the constellations points of the geometrically spacedsymbol constellation are variable depending on the channel SNR.
 19. Thecommunication system of claim 18, wherein: a backchannel communicationlink exists between the transmitter and the receiver; the receiver isconfigured to transmit information to the transmitter from which thetransmitter can compute the channel SNR; the transmitter is configuredto transmit to the receiver information indicative of a symbolconstellation and a code configuration; and the demapper is configuredto move the constellation points of the geometrically shaped symbolconstellation based upon the information received from the transmitter.20. A method of transmitting bits of user information between atransmitter and a receiver over a communication channel, comprising:using a coder within the transmitter to encode the bits of userinformation in accordance with a coding scheme; using a mapper withinthe transmitter to map the encoded bits of user information to a symbolconstellation, wherein the symbol constellation is a capacity optimizedgeometrically spaced symbol constellation that provides a given capacityat a reduced signal-to-noise ratio compared to a signal constellationthat maximizes d_(min); using a modulator within the transmitter tomodulate the symbols in accordance with a modulation scheme;transmitting the modulated signal via the communication channel usingthe transmitter; receiving a modulated signal at the receiver; using ademodulator within the receiver to demodulate the modulated signal inaccordance with the modulation scheme; using a demapper within thereceiver to demap the demodulated signal using the geometrically shapedsignal constellation to produce likelihoods; and decoding thelikelihoods using the receiver to obtain an estimate of the decodedbits.
 21. The method of claim 20, wherein the geometrically shapedsymbol constellation is optimized for capacity using parallel decodecapacity.
 22. The method of claim 20, wherein the geometrically shapedsymbol constellation is optimized for capacity using joint capacity. 23.A method of transmitting bits of user information between a transmitterand a receiver over a communication channel, comprising: transmittinginformation over a channel using a geometrically shaped symbolconstellation; determining the locations of points within a symbolconstellation having a fixed number of constellation point to achieve aninitial target user data rate; configuring the transmitter and thereceiver using the symbol constellation; using the transmitter to:encode the bits of user information in accordance with a coding scheme;map the encoded bits of user information to the symbol constellation;modulate the symbols in accordance with a modulation scheme; transmitthe modulated signal via the communication channel: using the receiverto: receive modulated signals transmitted by the transmitter; demodulatethe modulated signal in accordance with the modulation scheme; demap thedemodulated signal using the signal constellation to producelikelihoods; decode the likelihoods to obtain an estimate of the decodedbits; modifying the locations of the fixed number of points within thegeometrically shaped symbol constellation to change the target user datarate; and configuring the transmitter and the receiver with the modifiedsymbol constellation during the transmission of bits of userinformation.